Right-derivative

Table 3.10. Second derivatives, first left- and right-derivatives calculated for each data point listed in Table 3.9 through equations (3.1.43) to (3.1.46).

This erroneous statement comes about by confusing the discrete //-curve with its smoothened interpolating curve. For the latter the left and right derivatives would be equal at every point, which leads immediately to an equation similar to Eq. (51). However, such a conclusion cannot be transferred to difference quotients, which correspond to Eq. (51) itself. 

Top uiew of the possible excimer geometries and related photoproducts of meso-DlNEE (left) and rac-DlNEE (right) derived from molecular models... 

FIGURE 4 8 Oxidation of carbohydrate and fatty adds for the production of energy. In the resting state, glycogen in the liver is broken down to glucose units however, both liver (kft) and muscle (right) derive most of their energy from fatty acids released from adipose tissue into the bloodstrecim and oxidized via the Krebs cycle. When carbohydrate is oxidized in the liver, it tends to be released into the bloodstream as pyruvate and lactate. 

Strictly this is the right derivative. We can also define a left derivative... 

Fig. 6.15 Energy scheme for a metal and an adatom before (left) and after contact (right), derived according to the theoretical calculations given in refs. [20] and [21]...

Fig. 19. ng levels in 15 (left) and 16 (right) derived by a refined model with indication of the experimental values (Fig. 17)... 

Again the right derivative differs from the left derivative, as indicated by the sign. The maxima of/ indicate regions in the molecule, which prefer attack by a nucleophile, while/ exhibits maxima at sites susceptible to an attack by an electrophile. In other words,indicates where increase of electron density is energetically favorable, while/ is maximal where decrease of electron density is preferred. Practically, the Fukui functions are calculated by finite differences, e.g. ... 

Figure 8. The second fF", left) and fourth ( F"", right) derivatives of a spectrum consisting of two bands separated by D =...

Definition 2.1.1. A A-functor F J —> E is right-derivable if there exists a A-functor... 

Such a pair (RF, C) (respectively (LF, )) is called a right-derived (respectively left-derived) functor of F. 

Example 2.1.3. If F J —> E transforms quasi-isomorphisms into isomorphisms then F = Fog for a unique F Dj —> E and (F, identity) is both a right-derived and a left-derived functor of F. 

Corollary 2.2.7. Let A, A be abelian categories, let 3 C K(.4), J C K(.A ) be A-subcategories with canonical Junctors Q 3 —> Dj, Q 3 —> Dj/ to their respective derived categories, and let F 3 —> J and G J —> E be A-Junctors. Assume that G has a right-derived functor RG and that every complex X J admits a quasi-isomorphism into a right- Q F)-acydic complex Ax such that F Ax) is right-G-acyclic. Then Q F and GF have right-derived functors, denoted RF and R GF), and there is a unique A-Junctorial isomorphism... 

The basic example of a family (v x) in (2.2.6) arises when A has enough injectives, i.e., every object of A admits a monomorphism into an injective object. Then every complex X K (.A) admits a quasi-isomorphism iPx- X Ix into a bounded-below complex of injectives (see (1.8.2)) and by (2.3.4) and (2.3.2.1) below, this Ix is right-F-acyclic for every A-functor F K ( ) —> E, whence F is right-derivable. 

Corollary 2.3.2.3. Suppose that there exists a family of q-injective resolutions ipx Ix X J), i.e., for each X, is a quasi-isomorphism and Ix is 3-q-injective. Then any A-functor F J —> E has a right-derived functor (RF, C) with... 

Example 2.3.5. Let f/ be a topological space, O a sheaf of rings on U, and A the abelian category of left O-modules. Then a theorem of Spaltenstein [Sp, p. 138, Theorem 4.5] asserts that every complex in K( 4) admits a q-injective resolution. Hence by (2.3.2.3), every A-functor out of K( ) is right-derivable. 

The third is to prove the existence of the right adjoint of R p,) for (componentwise) proper morphism p, of diagrams of schemes. This is not so difficult, and is done in Chapter 17. We use Neeman s existence theorem on the right adjoint of triangulated functors. Not only to utilize Neeman s theorem, but to calculate composites of various left and right derived functors. 

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